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In this paper, a surface integral equation formulation for the prediction of corrosion-related electrostatic fields of conducting structures in an electrolyte is presented. The integral equation is discretized using the Locally Corrected Nystrom method. An iterative technique based on the Newton-Raphson method is described for use with nonlinear boundary conditions, and a Schur complement method is discussed to help improve the nonlinear solution efficiency. An image plane method is applied when an appropriate planar electrolyte-insulating bounding surface is present. The methods are validated by analysis of a canonical problem with analytic solution and by comparison to literature data. Higher-order solution convergence is observed for the canonical problem results.

A simple method for generating highorder, 3D surface meshes is presented. The higher-order meshes are suitable for investigating higherorder convergence in electromagnetic analysis codes. The mesh-generation algorithm uses two linear meshes: a coarse mesh and a refinement of the coarse mesh. The method is applied to both quadrilateral and triangle surface meshes. Mesh generation is validated by computing the discretization error for various meshes. Finally, integral equation error convergence of plane wave scattering from a conducting sphere is investigated in terms of mesh order and basis order. A reference implementation of the algorithms is made available.

In this paper, a technique to convert a locally corrected Nystrom discretization into a moment method discretization is detailed. The method is general and does not rely directly on any specific relationship between the basis functions used to compute the local corrections and the basis and test functions used in the moment method. The technique is applied to a quasi-magnetostatic volume integral equation for both interpolatory and algebraically constrained moment method bases. Numerical results for a variety of problems in terms of error convergence, system condition number, and degrees-of-freedom are provided.

In this paper, a nonlinear electrostatic surface integral equation is presented that is suitable for predicting corrosion-related fields. Nonlinear behavior arises due to electrochemical reactions at polarized surfaces. Hierarchical $H^{2}$ matrices are used to compress the discretized integral equation for the fast solution of large problems. A technique based on randomized linear algebra is discussed for the efficient computation of the Jacobian matrix required at each iteration of a nonlinear solution.

This communication presents an algebraic method for constructing arbitrary-order, divergence-conforming basis functions on hexahedral volume elements. The appropriate constraints within and on the boundaries of elements are provided. In particular, the handling of faces where quantities are discontinuous is discussed. The resulting bases are numerically characterized in terms of error convergence and system conditioning for a moment method discretization of the electric field volume integral equation for dielectric scatterers. Results show the accuracy of the proposed method as well as the low system matrix condition number that can be maintained as the basis order and mesh discretization are increased.

$H^{2}$ matrices provide compressed representations of the matrices obtained when discretizing surface and volume integral equations. The memory costs associated with storing $H^{2}$ matrices for static and low-frequency applications are $O(N)$ . However, when the $H^{2}$ representation is constructed using sparse samples of the underlying matrix, the translation matrices in the $H^{2}$ representation do not preserve any translational invariance present in the underlying kernel. In some cases, this can result in an $H^{2}$ representation with relatively large memory requirements. This paper outlines a method to compress an existing $H^{2}$ matrix by constructing a translationally invariant $H^{2}$ matrix from it. Numerical examples demonstrate that the resulting representation can provide significant memory savings.

This paper presents an algebraic method of generating arbitrary-order basis functions suitable for use in moment methods. The basis functions are constructed by enforcing a set of suitable constraints on a more general set of functions, leading to a matrix representation of the constraint condition. The desired bases are then found through use of the singular value decomposition. Divergence-conforming constrained bases are presented for quadrilateral meshes for use in electric and magnetic field integral equations. Appropriate constraints for enforcing normal continuity at cell boundaries are discussed. A variety of problems are analyzed using the constrained bases, and results show that the constrained bases exhibit exponential convergence for smooth structures and produce systems whose condition numbers grow relatively slowly with increasing basis order.

A surface integral equation method for computing the corrosion-related magnetostatic field is presented. An integral equation is solved for the corrosion-related electrostatic field in a homogeneous electrolyte from which the corrosion-related magnetostatic field is determined. The surface integral equation is discretized using the locally-corrected Nystrom method. The field analysis is validated through comparison to analytical and published results.

We report results obtained using a preconditioned fast iterative solver to model electrostatic circuit problems. The method of moments is used to discretize integral equation-based formulations of the underlying circuit problem. A compressed representation of the system matrix is obtained using the fast multipole method, and the resulting linear system is solved using a preconditioned iterative method. Preconditioners are constructed from a class of localization-based sparse direct solvers, and numerical performance is reported for realistic application ns. The impact of incorporating a multilevel matrix binormalization method is also examined, and resulting tradeoffs are discussed.

A high-order locally corrected Nystrom discretization for analyzing impressed current cathodic protection systems is presented. Non-linear polarization curves are incorporated using a Newton-Raphson scheme. A Schur complement scheme is introduced to handle large domains with small electrodes. The scheme is validated by comparing to the analytic solution for a sphere.

A new surface integral equation formulation for scattering from perfectly conducting objects is presented. The formulation is developed by adding a constraint on the normal component of the magnetic field to the augmented electric field integral equation (AEFIE) and extracting the static charge solution. The resulting AEFIEnH-S formulation is discretized using the method of moments with Rao–Wilton–Glisson (RWG) source functions and Buffa–Christiansen (BC) test functions. An iterative diagonal matrix scaling algorithm is used to improve the conditioning of the discrete system. Numerical examples demonstrate that the AEFIEnH-S is stable and accurate as the frequency is reduced for closed, open, and multiscale multiply connected geometries. The formulation relies only on diagonal preconditioning, it accurately models the near electric, near magnetic, and far fields, it does not require frequency scaling of the unknowns, and it does not incorporate any type of Helmholtz decomposition.

A generalization of the locally corrected Nystrom (LCN) discretization method is outlined wherein sparse transformations of the LCN system matrix are obtained via singular- value decompositions of local constraint matrices. The local constraint matrices are used to impose normal continuity of the currents across boundaries shared by mesh elements. Due to the method’s simplicity and flexibility, it is straightforward to develop high-order constrained LCN (CLCN) systems for different formulations and mesh element types. Numerical examples demonstrate the memory savings provided by the CLCN method and its improved accuracy when applied to geometries with sharp edges. It is also shown that the CLCN method maintains the high-order convergence of the LCN method, and it eliminates the need to include line charges in Nystrom-based discretizations of formulations that involve the continuity equation.

A well-conditioned, high-order Nystrom discretization of a volume integral equation for quasi-magnetostatic, time-harmonic eddy current analysis of magnetic-conducting materials is presented. Good conditioning is contingent on augmenting the system with a charge neutrality constraint. Two forms of the continuity constraint are discussed. An iterative diagonal preconditioning scheme is applied to greatly improve the condition number. Results are presented for canonical problems and the TEAM Workshop Problem 7.

A high-order Nystrom discretization of a volume integral equation derived for quasi-magnetostatic, time-harmonic eddy current analysis is presented. The formulation is based on the coupled magnetic field and magnetic vector potential integral equations. Both magnetic and conducting materials are included in the formulation. Analytic results for canonical problems are used to corroborate the high-order convergence of the method. Data for the TEAM Workshop problem 7 are presented.

This paper reports on an improved factorization strategy for integral equation matrices based on source and field transformations that have localizing radiation and reception properties. In addition to the use of localizing functions, the method incorporates orthogonally matched receivers. It is shown that this yields a controllably diagonal multilevel matrix factorization. To simplify the presentation, the algorithm is described for symmetric systems. Numerical results are presented for both symmetric and non-symmetric matrices.

In this paper, a technique for automatically generating basis functions suitable for integral equation discretizations is presented. The technique seeks to obtain an optimal basis function from a linear combination of more general functions by enforcing an appropriate set of constraints on the linear combination and then solving for the optimal coefficients. Numerical properties of divergence-conforming constrained bases on quadrilateral meshes are characterized in terms of scattering from perfectly electric conducting structures. Computed data show that the condition numbers of system matrices resulting from the use of the constrained bases are equal to or better than those resulting from the use of other basis sets reported in the literature.

Quasi-mixed order prism basis functions for use in Nystrom discretizations of volume integral equations are presented. The prism basis function set is easily mixed with other basis function sets such as those defined on hexahedra and tetrahedra. The prism basis functions are applied to solve a magnetostatic volume integral equation for the analysis of magnetic materials. Analytic results for canonical problems are used to corroborate the basis function set and convergence and error data are discussed.

A locally corrected Nystrom discretization of the magnetostatic volume integral equation is derived for the analysis of magnetic materials. The integral equation formulation incorporates both higher-order meshes and higher-order basis functions. A set of arbitrary order, hexahedral basis functions are presented. The formulation is applied to a set of canonical problems as well as TEAM Workshop problem number 13. Error convergence with respect to basis function order, mesh density, and mesh order is investigated, and results corroborate the formulation.

A method to analyze a one-dimensional array comprising tilted edge slots cut in the narrow wall of a rectangular waveguide is presented. The fields in the slots are calculated from a hybrid finite element-boundary integral (FE-BI) equation method. The boundary surface integrals on both the interior and exterior surface of the slots are computed from a spectrum of two-dimensional solutions (S2DS), which allows the influence of both the interior and exterior structure of the waveguide to be included rigorously. For initial array design, the mutual coupling between slots in the exterior region is treated approximately with an infinite array assumption modeled by two slots surrounded by periodic boundary walls with an imposed linear phase progression. A short, finite slot array is designed based on the infinite array assumption and fabricated. Results from full-wave analysis of the array and measurements are presented

A delta gap source formulation suitable for use in locally corrected Nystrom (LCN) discretizations of electric field integral equations is presented. The delta-gap is applied to an edge (or group of edges) between a set of faces. Due to the point-sampling used in the LCN method, the delta gap effect is distributed across the faces shared by the source edges. The LCN delta gap is validated using results from the moment method.

A constrained locally corrected Nystrom (CLCN) method has recently been developed. The CLCN method enables the imposition of normal continuity on underlying vector quantities across mesh element boundaries. This is accomplished by deriving appropriate transformation vectors through simple algebraic analyses of local, homogeneous constraint conditions. Compared to the LCN method, the CLCN method improves the condition numbers of the system matrix, it reduces computational costs through a reduction in the number of degrees of freedom (DOFs), and it improves accuracy for sharp geometries. In the case of the magnetic field integral equation (MFIE), it is shown that the CLCN method yields stable condition numbers as the order increases.

The electric field integral equation (EFIE) and the magnetic field integral equation (MFIE) suffer from a spurious interior resonance problem for closed structures. The linear combination of the two equations can remedy this issue, and the resulting formulation is referred as the combined field integral equation (CFIE). Due to its incorporation of the EFIE, the CFIE inherits a low-frequency breakdown problem. This breakdown is manifest through a poorly conditioned system matrix and poor accuracy for some problems. The CFIE also exhibits mesh instability due to the hypersingular EFIE.

A technique is presented to efficiently solve for the currents on an appendage, e.g., antenna, attached to a planar surface of a conducting body. The appendage may be embedded in a homogeneous, dielectric material. The technique presented alleviates the complications associated with the point where the appendage is attached to the body. To illustrate the method, a wire antenna attached to an axisymmetric body is analyzed in detail. A set of coupled integral equations are formulated, appropriate quantities are expanded into Fourier modes, and coupled integral equations are derived for the Fourier coefficients of the unknowns. These equations are solved and the input admittance of the wire antenna is determined from the computed currents and is corroborated by measurements.

A technique is described for determining the field in a series of cascaded, axisymmetric cylindrical cavities excited by a /spl phi/-independent source. The constituent cavities are axisymmetric but may have a cross-section that varies with axial displacement. A set of coupled integral equations are solved for the unknown electric fields in the apertures that separate the constituent cavities. Separate integral equations are formulated to determine the field in each cavity with a variable cross-section.

The inductance of a coil in a slotted, conducting shell is investigated. At frequencies for which the coil is very small relative to wavelength, an integral equation is formulated and numerically solved for the electric field induced in the slots. From knowledge of the slot electric field, the inductance of the coil in the slotted shell can be calculated. The dependence of inductance on the presence of the slotted shell is assessed quantitatively and explained on the basis of physical principles. The inductance of a one-turn loop inside a slotted shell is measured, and the data are found to be in excellent agreement with computations. As an unslotted shield closes on a coil, inductance is reduced noticeably, but if the shield possesses axial slots, the reduction is much less significant.

Integral equation methods provide an effective approach for solving electromagnetic radiation and scattering problems. For linear homogeneous materials, surface integral equations are commonly applied. However, for nonlinear, high contrast and/or inhomogeneous materials, volume integral equations (VIEs) often provide a more useful alternative. The locally corrected Nystrom (LCN) method provides a scheme for discretizing the VIE formulation. The main advantages of the LCN method over other discretization methods include the ability to easily handle different mesh element types at the same time, and the ability to easily incorporate high-order representations. Moreover, by applying the LCN discretization to the VIE formulation, the material properties can be confined to the diagonal of the system matrix, which significantly reduces the computational costs in cases for which a given geometry with different material properties is of interest. Unfortunately, the regular VIE formulation has a number of limitations. These limitations include poor matrix condition numbers for problems with high contrast materials, and deteriorating performance for problems with complex, multi-scale meshes.

A quasi-magnetostatic volume integral equation discretized using the Nystrom method is presented. The integral equation is formulated in terms of the differential susceptibility for use in hysteresis modeling. A simple stepped algorithm for hysteresis modeling of non-linear magnetic materials is studied. The algorithm is investigated using two non-linear models: a unified hysteresis model and the Jiles-Atherton model. Results are presented for a magnetic sphere excited by an alternating field and the computed results compare well with analytic results. The algorithm also captures minor loops.

The focus of this chapter is to provide a broad overview of the high-order Locally Corrected Nystrom (LCN) method. The LCN method is applied herein to the solution of a general hybrid surface/volume integral equation defining the electromagnetic scattering and/or radiation from arbitrary three-dimensional geometries of arbitrary material composition. The chapter starts out by providing a basic understanding of the LCN algorithm and its analogy to the more popular method of moments. The LCN method is then specialized to the solution of hybrid integral equations by providing nuts-and-bolts details regarding the application of the method. LCN discretizations of surface integral equations are presented for curvilinear quadrilateral or triangular meshes. Furthermore, LCN discretizations of volume integral equations are derived for curvilinear hexahedral, tetrahedral, or prism meshes. Quadrature rules suitable for a Nystrom discretization and basis functions suitable for local corrections are defined for each topological type. Performing the local corrections for the different kernels encountered is also discussed. Finally a number of numerical examples are presented that span the different integral operators and topological types. For all cases, data is provided that demonstrates the high-order convergence of a LCN solution method. It is hoped that the power, yet simplicity, of the LCN method is conveyed in this chapter, along with enough details for an interested reader to develop their own LCN software.

A time-harmonic quasi-magnetostatic volume integral equation is formulated for the eddy currents and magnetization in a conducting, magnetic material. The integral equation is solved using the Locally Corrected Nystrom (LCN) method. The formulation is capable of handling mixed meshes of hexahedral, prism, and tetrahedral elements. Several problems are analyzed and error convergence plots are presented.

Local-global solution (LOGOS) modes have been proven to be an effective framework for developing fast, direct solution methods for electromagnetic simulations [1]. The LOGOS method provides an efficient direct factorization that sparsifies dense linear systems of equations with controlled accuracy. In this paper, the LOGOS method is applied to analyze the magnetostatic problem. The analysis of magnetic materials in a magnetostatic field is a challenging problem. To add to the challenge, magnetic materials are usually non-linear and are often inhomogeneous. An efficient solution is proposed for this class of problems that is based on a magnetostatic volume integral equation (MVIE) discretized via a Locally Corrected Nystrom method [2, 3] combined with a fast, direct LOGOS solver. The MVIE is posed such that only the diagonal operator is influenced by the magnetic material parameters. The dense linear system can be pre-factored via a LOGOS factorization. Thus, each iteration of a non-linear solution does not require the system matrix to be re-factorized and the problem can be solved in O(N) operations. In this paper, the Locally Corrected Nystrom solution of the proposed MVIE and the efficiency of the fast direct LOGOS solution are validated.

A method is presented in this paper to rigorously analyze and design a linear edge slot array with appendages attached to the external structure. The analysis method is a finite element boundary integral (FE-BI) method that uses a spectrum of two-dimensional solutions to compute the boundary integral terms. Full array optimization is performed by iteration to obtain slot dimensions that produce the desired aperture distribution. Since the optimization attempts to control the slot electric fields directly, side lobe levels within a few dB of ideal results are obtained. A flare with a choke is proposed to control the vertical beamwidth and cross-polarization. Results for an optimized 75-slot array are given and discussed and good corroboration of the data is obtained with Ansoft's HFSS

A linear array antenna comprised of tilted edge slots cut in the narrow wall of a rectangular waveguide is analyzed. The analysis method is a hybrid FEM/BIE method that uses spectral domain integral equations to determine the boundary elements in the FEM analysis. The mutual coupling effects in a large array are approximated with an infinite array assumption. Calculations are compared to measurements as well as results from Ansoft's HFSS simulator.

A method is described to accurately analyze horizontal and vertical slots cut in the wall of a cylindrical waveguide. The interior and exterior regions of the guide are analyzed by using a spectrum of two-dimensional solutions (S2DS method). In addition, the finite thickness of the waveguide wall is also taken into account. To accelerate the convergence of the spectral integrals in the S2DS method, the asymptotic behavior of the integrands in these integrals is identified. The reflection and transmission coefficients for horizontal and vertical, rectangular slots in the broad wall of a rectangular waveguide are computed.